The modern definition of (closed) model category differs in two ways from Quillen's 1969 definition:

  • Model categories are now required to be complete and cocomplete, whereas Quillen only asked for finite limits and finite colimits.

  • The two factorisation systems are now required to be functorial.

These changes do make the two definitions genuinely different, since there are non-trivial small model categories in the old sense but not in the new sense.

Question. Are there any textbook results about model categories in the modern sense that are invalid for model categories in the old sense?

I ask because the majority of the textbooks I have looked at so far (e.g. [Hovey, 1999], [Hirschhorn, 2003], [Dwyer, Hirschhorn, Kan, and Smith, 2004]) use the stronger definition. While propositions like "There exists a functorial choice of fibrant/cofibrant replacement" obviously depend on functorial factorisation, it is not so easy to decide whether propositions like "A left adjoint is a left Quillen functor if and only if it preserves cofibrations between cofibrant objects and all trivial cofibrations" [Hirschhorn, 2003, Prop. 8.5.4] needs the stronger definition.

Bonus question. When did people start requiring functorial factorisations, and who if anyone started the trend?

I notice that [Dwyer and Kan, 1980, Function complexes in homotopical algebra] mention functorial factorisation as a nice optional extra, so the idea goes back at least that far.

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    $\begingroup$ Assuming bicompleteness seems to be quite standard. On factorizations May & Ponto "More concise algebraic topology" write The parenthetical (functorial) in the definition is a matter of choice, depending on taste and convenience. Quillen’s original definition did not require it, but many more recent sources do. There are interesting model categories for which the factorizations cannot be chosen to be functorial (see for example [68]), but they can be so chosen in the examples that are most commonly used. We will not go far enough into the theory for the difference to matter significantly. $\endgroup$ – Martin Feb 11 '13 at 2:02
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    $\begingroup$ [68] is: Isaksen, A model structure on the category of pro-simplicial sets. // A reference that does not require functorial factorizations is Goerss-Jardine. $\endgroup$ – Martin Feb 11 '13 at 2:07
  • $\begingroup$ I believe this should be tagged reference-request. I'm not sure if it's right, though. $\endgroup$ – user122283 Aug 31 '14 at 16:52
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    $\begingroup$ No. I'm not looking for a reference. $\endgroup$ – Zhen Lin Aug 31 '14 at 17:18
  • $\begingroup$ I suggest you read the paper by Dywer and Splaniski on Model Categories. I believe it sticks to the Quillen definition of Model Categories. $\endgroup$ – user417289 Nov 5 '17 at 21:44

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